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classes:2009:fall:phys4101.001:lec_notes_1207 [2009/12/08 10:41] – myersclasses:2009:fall:phys4101.001:lec_notes_1207 [2009/12/15 23:58] (current) – ely
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 ===== Dec 07 (Mon)  ===== ===== Dec 07 (Mon)  =====
-** Responsible party: John Galt, Dark Helmet ** +** Responsible party: John Galt, Dark Helmet, Esquire ** 
  
 **To go back to the lecture note list, click [[lec_notes]]**\\ **To go back to the lecture note list, click [[lec_notes]]**\\
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 ====Chapter 6: Time Indepent Perturbation Theory==== ====Chapter 6: Time Indepent Perturbation Theory====
 +
 +We do it becuase it is a useful tool
 +
 +Shroedinger's equation is the must fundamental tool for QM.
 +We take solutions and eigenstates/eigenvectors to get energy levels
  
 ===Non-Degenerate Case=== ===Non-Degenerate Case===
 +This is the simplest case
 +single energy->single equation
  
 <math> H_0|\psi_n^{(0)}>=E_n^{(0)}|\psi_n^{(0)}> </math> <math> H_0|\psi_n^{(0)}>=E_n^{(0)}|\psi_n^{(0)}> </math>
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 A Fourier expansion can be used to express <math> |\psi_n^{(1)}>=\Sigma C_{mn}|\psi_n^{(0)}> </math> where m≠n A Fourier expansion can be used to express <math> |\psi_n^{(1)}>=\Sigma C_{mn}|\psi_n^{(0)}> </math> where m≠n
  
-Plugging this into the new Hamiltion +Plugging this into the new Hamiltion yields 
 + 
 +<math> (H_0+\lambda H')(|\psi_n^{(0)}>+\lambda|\psi_n^{(1)}>)=(E_n^{(0)}+\lambda E_n^{(1)})(|\psi_n^{(0)}>+\lambda|\psi_n^(1)>) </math> 
 + 
 +<math> H_0|\psi_n^{(0)}>+\lambda(H'|\psi_n^{(0)}>+H_0|\psi_n^{(1)}>)=E_n^{(0)}|\psi_n^{(0)}>+\lambda(E_n^{1}|\psi_n^{(0)}>+E_n^{(0)}|\psi_n^{(1)}>) </math> 
 + 
 +<math> H'|\psi_n^{(0)}>+H_0|\psi_n^{(1)}>=E_n^{(1)}|\psi_n^{(0)}>+E_n^{(0)}|\psi_n^{(1)}> </math> 
 + 
 +Now using the Fourier expansion expression 
 + 
 +<math> H'|\psi_n^{(0)}>+\Sigma C_{nm}H_0|\psi_m^{(0)}>=E_n^{(1)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}|\psi_m^{(0)}>  </math> 
 + 
 +<math> H'|\psi_n^{(0)}+\Sigma C_{nm}E_m^{(0)}|\psi_m^{(0)}>=E_n^{(1)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}|\psi_m^{(0)}> </math> 
 + 
 +Using this, one can find an expression for the expectation of the new Hamiltonian as follows 
 + 
 +<math> <\psi_n^{(0)|H'|\psi_n^{(0)}>+\Sigma C_{nm}E_m^{(0)}<\psi_n^{(0)}|\psi_m^{(0)}>=E_n^{(1)}<\psi_n^{(0)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}<\psi_n^{(0)}|\psi_m^{(0)}> </math> 
 + 
 +<math><\psi_n^{(0)|H'|\psi_n^{(0)}>=E_n^{(1)} </math> 
 + 
 +Now one can introduce a new parameter l≠n but l can equal m and show 
 + 
 +<math><\psi_l^{(0)|H'|\psi_n^{(0)}>+\Sigma C_{nm}E_m^{(0)}<\psi_l^{(0)}|\psi_m^{(0)}>=E_n^{(1)}<\psi_l^{(0)}|\psi_n^{(0)}>+E_n^{(0)}\Sigma C_{nm}<\psi_l^{(0)}|\psi_m^{(0)}> </math> 
 + 
 +<math><\psi_l^{(0)|H'|\psi_n^{(0)}>+C_{nl}E_l^{(0)}=C_{nl}E_n^{(0)}  </math> 
 + 
 +⇒<math>C_{nl}=<\psi_l^{(0)|H'|\psi_n^{(0)}>/(E_n^{(0)}-E_l^{(0)}) </math> 
 + 
 +⇒<math>E_n^{(2)}=\Sigma|<\psi_l^{(0)|H'|\psi_n^{(0)}>|^2/(E_n^{(0)}-E_l^{(0)}) </math> 
 + 
 +This was all i had for notes as well-Dark Helmet
  
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classes/2009/fall/phys4101.001/lec_notes_1207.1260290465.txt.gz · Last modified: 2009/12/08 10:41 (external edit)

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